Chapter 2

Technology. Let \(F(x)=\sqrt{x}\). The polynomials $$P_{2}(x)=\frac{3}{4}+\frac{3}{8} x-\frac{1}{64} x^{2} \quad \text { and } \quad P_{3}(x)=\frac{5}{8}+\frac{15}{32} x-\frac{5}{128} x^{2}+\frac{1}{512} x^{3}$$ closely approximate \(F\) near the point (4,2) of \(F\). a. Draw the graphs of \(F\) and \(P_{2}\) on the range \(1 \leq x \leq 8\). b. Compute the relative error in \(P_{2}(2)\) as an approximation to \(F(2)=\sqrt{2}\). c. Draw the graphs of \(F\) and \(P_{3}(x)\) on the range \(1 \leq x \leq 8\). d. Compute the relative error in \(P_{3}(2)\) as an approximation to \(F(2)=\sqrt{2}\).

Let \(\mathrm{F}\) be the collection of ordered number pairs to which an ordered pair \((x, y)\) belongs if and only if \(x\) is a number and \(y=x^{2}+x\). a. Which of the ordered number pairs belong to F? (0,1),(0,0),(1,1),(1,3),(1,-1),(-1,1) , (-1,0),(-1,-1) b. Is there any uncertainty as to the members of \(\mathrm{F} ?\) c. What is the domain of \(\mathrm{F} ?\) d. What is the range of \(\mathrm{F} ?\)

Technology. Let \(F(x)=\sqrt[3]{x}\). The polynomials $$P_{2}(x)=\frac{5}{9}+\frac{5}{9} x-\frac{1}{9} x^{2} \quad \text { and } \quad P_{3}(x)=P_{2}(x)+\frac{5}{81}(x-1)^{3}$$ closely approximate \(F\) near the point (1,1) of \(F\). a. Draw the graphs of \(F\) and \(P_{2}\) on the range \(0 \leq x \leq 3\). b. Compute the relative error in \(P_{2}(2)\) as an approximation to \(F(2)=\sqrt[3]{2}\). c. Draw the graphs of \(F\) and \(P_{3}\) on the range \(1 \leq x \leq 3\). d. Compute the relative error in \(P_{3}(2)\) as an approximation to \(F(2)=\sqrt[3]{2}\).

For the function, \(F,\) defined by \(F(x)=x^{2},\) 1\. Compute \(F(1+2)\), and \(F(1)+F(2)\). Is \(F(1+2)=F(1)+F(2)\) ? 2\. Compute \(F(3+5),\) and \(F(3)+F(5) .\) Is \(F(3+5)=F(3)+F(5) ?\) 3\. Compute \(F(0+4),\) and \(F(0)+F(4) .\) Is \(F(0+4)=F(0)+F(4) ?\)

For the following experiments, determine the independent variable and the dependent variable, and draw a simple graph or give a brief verbal description (your best guess) of the function relating the two. a. A rabbit population size is a function of the number of coyotes in the region. b. An agronomist, interested in the most economical rate of nitrogen application to corn, measures the corn yield in test plots using eight different levels of nitrogen application. c. An enzyme, E, catalyzes a reaction converting a substrate, \(\mathrm{S},\) to a product \(\mathrm{P}\) according to $$ \mathrm{E}+\mathrm{S} \rightleftharpoons \mathrm{ES} \rightleftharpoons \mathrm{E}+\mathrm{P} $$ Assume enzyme concentration, [E], is fixed. A scientist measures the rate at which the product P accumulates at different concentrations, [S], of substrate. d. A scientist titrates a \(0.1 \mathrm{M}\) solution of \(\mathrm{HCl}\) into \(5 \mathrm{ml}\) of an unknown basic solution containing litmus (litmus causes the color of the solution to change as the pH changes).

Suppose your are traveling an interstate highway and that every 10 miles there is an emergency telephone. Let \(D\) be the function defined by $$ D(x) $$ is the distance to the nearest emergency telephone where \(x\) is the mileage position on the highway. a. Draw a graph of \(D\). b. Find the period and amplitude of \(D\).

Technology Shown in the Table 2.4 are the densities of water at temperatures from 0 to \(100^{\circ} \mathrm{C}\) Use your calculator or computer to fit a cubic polynomial to the data. See Explore 2.5 .1 and Explore 2.5 .2 . Compare the graphs of the data and of the cubic.Table 2.4: The density of water at various temperatures Source: Robert C. Weast, Melvin J. Astle, and William H. Beyer, CRC Handbook of Chemistry and Physics, 68 th Edition, 1988 , CRC Press, Boca Raton, FL, p F-10. \(D(T)=1.00004105+0.00001627 T-0.000005850 T^{2}+0.000000015324 T^{3}\)

Find a function, L, defined for all numbers (domain is all numbers) such that for all numbers a and \(\mathrm{b}, \mathrm{L}(\mathrm{a}+\mathrm{b})=\mathrm{L}(\mathrm{a})+\mathrm{L}(\mathrm{b})\). Is there another such function?

Your 26 inch diameter bicycle wheel has a patch on it. Let \(P\) be the function defined by \(P(x) \quad\) is the distance from patch to the ground where \(x\) is the distance you have traveled on a bicycle trail. a. Draw a graph of \(P\) (approximate is acceptable). b. Find the period and amplitude of \(P\).

Which of the following functions are invertible? a. The distance a DNA molecule will migrate during agarose gel electrophoresis as a function of the molecular weight of the molecule, for domain: \(1 \mathrm{~kb} \leq\) Number of bases \(\leq 20 \mathrm{~kb}\). b. The density of water as a function of temperature. c. Day length as a function of elevation of the sun above the horizon (at, say, 40 degrees North latitude). d. Day length as a function of day of the year.

Find functions, \(F(z)\) and \(G(x)\) so that the following functions, \(H,\) may be written as \(F(G(x))\). $$ \begin{array}{lll} \text { a. } H(x) & =\left(1+x^{2}\right)^{3} & \text { b. } H(x)=10^{\sqrt{x}} & \text { c. } H(x)=\log \left(2 x^{2}+1\right) \\ \text { d. } H(x) & =\sqrt{x^{3}+1} & \text { e. } H(x)=\frac{1-x^{2}}{1+x^{2}} & \text { f. } H(x)=\log _{2}\left(2^{x}\right) \end{array} $$

For the function, \(F(x)=x^{2}+x,\) compute the following (a) \(\frac{F(5)-F(3)}{5-3}\) (b) \(\frac{F(3+2)-F(3)}{2}\) (c) \(\frac{F(b)-F(a)}{b-a}\) (d) \(\frac{F(a+h)-F(a)}{h}\)

Let \(f\) be the function defined by $$ f(x)=1-x^{2} \quad-1 \leq x \leq 1 $$ Let \(F\) be the extension of \(f\) with period 2 . a. Draw a graph of \(f\). b. Draw a graph of \(F\). c. Evaluate \(F(1), F(2), F(3), F(12), F(31)\) and \(F(1002)\). d. Find the amplitude of \(F\).

Find functions, \(F(u)\) and \(G(v)\) and \(H(x)\) so that the following functions, \(K,\) may be written as \(F(G(H(x)))\). $$ \begin{array}{lll} \text { a. } K(x) & =\sqrt{1-\sqrt{x}} & \text { b. } K(x)=\left(1+2^{x}\right)^{3} & \text { c. } K(x) & =\log \left(2 x^{2}+1\right) \\ \text { d. } K(x) & =\sqrt{x^{3}+1} & \text { e. } K(x)=\left(1-2^{x}\right)^{3} & \text { f. } K(x) & =\log _{2}\left(1+2^{x}\right) \end{array} $$

Shown in Figure Ex. 2.6 .6 is the graph of \(F(x)=2^{x} .(-2,1 / 4),(0,1),\) and (2,4) are ordered pairs of \(F\). What are the corresponding ordered pairs of \(F^{-1}\) ? Plot those points and draw the graph of \(F^{-1}\).

Find the periods of the following functions. \(\begin{array}{ll}\text { a. } & P(t)=\sin \left(\frac{\pi}{3} t\right) \\\ \text { b. } & P(t)=\sin (t) \\ \text { c. } & P(t)=5-2 \sin (t)\end{array}\) \(\begin{array}{ll}\text { d. } & P(t)=\sin (t)+\cos (t) \\ \text { e. } & P(t)=\sin \left(\frac{2 \pi}{2} t\right)+\sin \left(\frac{2 \pi}{3} t\right) \\\ \text { f. } & P(t)=\tan 2 t\end{array}\)

Compute the compositions, \(f(g(x))\), of the following pairs of functions. In each case specify the domain and range of the composite function, and sketch the graph. Your calculator may assist you. For example, the graph of part A can be drawn on the TI- 86 calculator with GRAPH \(, \quad y(x)=, \quad y 1=x^{-2}, \quad y 2=1 /(1+y 1)\) You may wish to suppress the display of \(y 1\) with SELCT in the \(\mathrm{y}(\mathrm{x})=\) menu. a. \(\begin{array}{lllll}f(z) & =\frac{1}{1+z} & \text { b. } & f(z)=\frac{z}{1+z} & \text { c. } & f(z)=5^{z} \\ g(x) & =x^{2} & & g(x)=\frac{x}{1-x} & & g(x)=\log x\end{array}\) \(\begin{array}{llllll}\text { d. } \quad f(z) & =\frac{1}{z} & \text { e. } & f(z)=\frac{z}{1-z} & \text { f. } & f(z)=\log z \\ g(x) & =1+x^{2} & & g(x)=\frac{x}{1+x} & & g(x)=5^{x}\end{array}\) \(\begin{array}{llllll}\text { g. } f(z) & =2^{z} & \text { h. } & f(z)=2^{z} & \text { i. } & f(z)=\log z \\ g(x) & =-x^{2} & & g(x)=-1 / x^{2} & & g(x)=1-x^{2}\end{array}\)

A bit of a difficult exercise. For any location, \(\lambda\) on Earth, let Annual Daytime at \(\lambda, A D(\lambda),\) be the sum of the lengths of time between sunrise and sunset at \(\lambda\) for all of the days of the year. Find a reasonable formula for \(A D(\lambda)\). You may guess or find data to suggest a reasonable formula, but we found proof of the validity of our formula a bit arduous. As often happens in mathematics, instead of solving the actual problem posed, we found it best to solve a 'nearby' problem that was more tractable. The \(365.24 \ldots\) days in a year is a distraction, the elliptical orbit of Earth is a downright hinderance, and the wobble of Earth on its axis can be overlooked. Specifically, we find it helpful to assume that there are precisely 366 days in the year (after all this was true about 7 or 8 million years ago), the Earth's orbit about the sun is a circle, the Earth's axis makes a constant angle with the plane of the orbit, and that the rays from the sun to Earth are parallel. We hope you enjoy the question.

Sketch the graphs and label the axes for $$\text { (a) } \quad y=0.2 \cos \left(\frac{2 \pi}{0.8} t\right) \quad \text { and } \quad \text { (b) } \quad y=5 \cos \left(\frac{1}{8} t+\pi / 6\right)$$

For each part, find two pairs, \(F\) and \(G,\) so that \(F \circ G\) is \(H\). a \(H(x)=\sqrt{1-\sqrt{x}}\) b \(H(x)=\frac{1}{1-\sqrt{x}} \quad\) c \(H(x)=\left(1+x^{2}\right)^{3}\) \(\mathrm{d} \quad H(x)=\left(x^{\left(x^{2}\right)}\right)^{3}\) e \(H(x)=2^{\left(x^{2}\right)}\) \(\mathrm{f} \quad H(x)=\left(2^{x}\right) 2\)

What are the implied domains of the functions $$ \begin{array}{ll} F(x)=\sqrt{x-1} & F(x)=\frac{1+x^{2}}{1-x^{2}} \\ F(x)=\sqrt{4-x^{2}} & F(x)=\log _{10}\left(x^{2}\right) \end{array} $$

Air is flowing into a spherical balloon at the rate of \(10 \mathrm{~cm}^{3} / \mathrm{s}\). What volume of air is in the balloon \(t\) seconds after there was no air in the balloon? The volume of a sphere of radius \(r\) is \(V=\frac{4}{3} \pi r^{3}\). What will be the radius of the balloon \(t\) seconds after there is no air in the balloon?

Is there an invertible function whose domain is the set of positive numbers and whose range is the set of non-negative numbers?

Why are all the points of the graph of \(y=\log _{10}(\sin (x))\) on or below the X-axis? Why are there no points of the graph with \(x\) -coordinates between \(\pi\) and \(2 \pi ?\)

Technology Draw the graph of the composition of \(F(x)=10^{x}\) with \(G(x)=\log _{10} x .\) Now draw the graph of the composition of \(\mathrm{G}\) with \(\mathrm{F}\). Explain the difference between the two graphs.

Answer the question in Explore 2.6.2, Suppose G is the inverse of an invertible function \(\mathrm{F}\). What is the inverse of \(\mathrm{G}\) ?

Let \(P(x)=2 x^{3}-7 x^{2}+5\) and \(Q(x)=x^{2}-x\). Use algebra to compute \(Q(P(x))\). You may conclude (correctly) from this exercise that the composition of two polynomials is always a polynomial.

Find equations for the inverses of the functions defined by (a) \(F_{1}(x)=\frac{1}{x+1}\) (b) \(\quad F_{2}(x)=\frac{x}{x+1}\) (c) \(\quad F_{3}(x)=1+2^{x}\) (d) \(\quad F_{4}(x)=\log _{2} x-\log _{2}(x+1)\) (e) \(\quad F_{5}(x)=10^{-x^{2}}\) for \(\mathrm{x} \geq 0\) (f) \(\quad F_{6}(z)=\frac{z+\frac{1}{z}}{2}\) for \(\mathrm{z} \geq 1\) (g) \(\quad F_{7}(x)=\frac{2^{x}-2^{-x}}{2}\)

Technology Draw the graphs of \(F(x)=\sin x\) and $$F(x)=\sin x \quad \text { and } \quad P_{5}(x)=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}$$ on the range \(0 \leq x \leq \pi\). Compute the relative error in \(P_{5}(\pi / 4)\) as an approximation to \(F(\pi / 4)\) and in \(P_{5}(\pi / 2)\) as an approximation to \(F(\pi / 2)\).

Polynomial approximations to the cosine function. a. Draw the graphs of \(F(x)=\cos x\) and $$ F(x)=\cos x $$ and $$ P_{2}(x)=1-\frac{x^{2}}{2} $$ on the range \(0 \leq x \leq \pi\). b. Compute the relative error in \(P_{2}(\pi / 4)\) as an approximation to \(F(\pi / 4)\) and in \(P_{2}(\pi / 2)\) as an approximation to \(F(\pi / 2)\) c. Use a graphing calculator to draw the graphs of \(F(x)=\cos x\) and $$ P_{4}(x)=1-\frac{x^{2}}{2}+\frac{x^{4}}{24} $$ on the range \(0 \leq x \leq \pi\). d. Compute the relative error in \(P_{4}(\pi / 4)\) as an approximation to \(F(\pi / 4)\) and the absolute error in \(P_{4}(\pi / 2)\) as an approximation to \(F(\pi / 2)\).